This result shows that the transverse current is a pure spin-current, in agreement with the results known from the direct resolution of the drift-diffusion equations[1]}, [2]}, [3]}, [4]}, [5]}. Indeed, the charge current vanishes \(J_y^Q \equiv J^0_{y \uparrow } + J^0_{y \downarrow } = 0\) , and the spin-current reads \(J^{Sp}_y \equiv J^0_{y \uparrow } - J^0_{y \downarrow } = 2 J^0_{y \uparrow }\) .
